Mathematics for teachers of mathematics

My new paper at The De Morgan Gazette:

A. Borovik, Mathematics for teachers of mathematics, The De Morgan Gazette 10 no. 2 (2018), 11-25. bit.ly/2NWECtn


The paper contains a sketch of a BSc Hons degree programme Mathematics (for Mathematics Education). It can be seen as a comment on Gardiner (2018) where he suggests that the current dire state of mathematics education in England cannot be improved without an improved structure for the preparation and training of mathematics teachers:

Effective preparation and training requires a limited number of national institutional units, linked as part of a national effort, and subject to central guidance. For recruitment and provision to be efficient and effective, each unit should deal with a significant number of students in each area of specialism (say 20–100). In most systems the initial period of preparation tends to be either

  •  a “degree programme” of 4–5 years (e.g. for primary teachers), with substantial subject-specific elements, or
  • an initial specialist, subject-based degree (of 3+ years), followed by (usually 2 years) of pedagogical and didactical training, with some school experience.

This paper suggests possible content, and didactic principles, of

a new kind of “initial specialist, subject-based degree” designed for intending teachers.

This text is only a proof of concept; most details are omitted; those that are given demonstrate, I hope, that a new degree would provide a fresh and vibrant approach to education of future teachers of mathematics.


UKRI: Accelerating the transition to full and immediate Open Access to scientific publications

Yesterday, 4 September 2018, UKRI announced their

Plan S: Accelerating the transition to full and immediate Open Access to scientific publications

Since the LMS critcally depends on income from publishing, it has serious implications for out Society.

The key principle of the Plan is as follows:

“After 1 January 2020 scientific publications on the results from research funded by public grants provided by national and European research councils and funding bodies, must be published in compliant Open Access Journals or on compliant Open Access Platforms.”


  • Authors retain copyright of their publication with no restrictions. All publications must be published under an open license, preferably the Creative Commons Attribution Licence CC BY. In all cases, the license applied should fulfil the requirements defined by the Berlin Declaration;
  • The Funders will ensure jointly the establishment of robust criteria and requirements for the services that compliant high quality Open Access journals and Open Access platforms must provide;
  • In case such high quality Open Access journals or platforms do not yet exist, the Funders will, in a coordinated way, provide incentives to establish and support them when appropriate; support will also be provided for Open Access infrastructures where necessary;
  • Where applicable, Open Access publication fees are covered by the Funders or universities, not by individual researchers; it is acknowledged that all scientists should be able to publish their work Open Access even if their institutions have limited means;
  • When Open Access publication fees are applied, their funding is standardised and capped (across Europe);
  • The Funders will ask universities, research organisations, and libraries to align their policies and strategies, notably to ensure transparency;
  • The above principles shall apply to all types of scholarly publications, but it is understood that the timeline to achieve Open Access for monographs and books may be longer than 1 January 2020;
  • The importance of open archives and repositories for hosting research outputs is acknowledged because of their long-term archiving function and their potential for editorial innovation;
  • The `hybrid’ model of publishing is not compliant with the above principles;
  • The Funders will monitor compliance and sanction non-compliance.

Tony Gardiner: Towards an effective national structure for teacher preparation and support in mathematics

A new paper at The De Morgan Gazette:

A. D. Gardiner, Towards an effective national structure for teacher preparation and support in mathematics, The De Morgan Gazette 10 no. 1 (2018), 1-10. bit.ly/2N9NU7W


The fragmented, learn-on-the-job English model for ITE is not working. About this there is little dispute.  We analyse why such a system cannot possibly work for mathematics teaching.  We also suggest the need for an improved national framework for teacher preparation and development, based on a limited number of specialist centres, which accumulate expertise over time, and through which planned programmes might be effectively delivered.

Trigger reflex

When I see a statement like that, I just cannot stop myself from pulling the trigger:

Why does Wittgenstein want surveyability? He seems to think that to be capable of the specific use of a theorem which a new proof makes possible we must be able  to reproduce its proof. This is just false, indeed perversely so — without understanding anything about Wiles’ proof of Fermat’s Last Theorem you can use it to rule out the truth of \(a^{17} + b^{17} = c^{17} \) where \(a\), \(b\) and \(c\) are any three integers, even hundreds of digits long — for example I know that \(123456789^{17} + 12233445566778899^{17}\) can’t be equal to \(12345678901234567890^{17}\) without needing to calculate any of the three powers. [Edwin Coleman, The surveyability of long proofs, Foundations of Science, 14, Issue 1–2pp 27–43.]

Indeed, I believe most mathematicians will make an instant observation that \(17\) is a odd natural number, and therefore the last digit of \(9^{17}\) is \(9\), and therefore the last digit of  \(123456789^{17} + 12233445566778899^{17}\) is \(8\) and does not equal to the last digit of  \(12345678901234567890^{17}\) , which is, of course, \(0\). One does not need Fermat’s Last Theorem for that (and, for the sake of historical integrity of the narrative, the case \(n = 17\) had been settled by Kummer in 1847).

Do mathematicians need funding?

My answer to a question on Quora: Do mathematicians need funding?

There was a number of answers in this thread regurgitating the old joke about mathematicians and philosophers – apparently the latter require even less funding. What really matters, however, is that mathematicians really need (very modest, admittedly) funding for meeting regularly other mathematicians – either at conferences, or, which is much more important, in direct visits to their universities, or in various programs like “Research in Pairs” run by the famous Mathematisches Forschungsinstitut Oberwolfach. There are serious psychological and neurophysiological reasons why doing serious mathematics in isolation from the community of fellow mathematicians is very difficult, next to impossible. I served, for a number of years, on a committee of a learned society which was awarding quick, easy to apply, small grants for international collaborative research in mathematics (basically, short visits, for about a week) . Quality of papers arising from this research is astonishingly high, on the value for money count it beats any state-funded bureaucratic schemes.


The effect of domestication on post-conflict management: wolves reconcile while dogs avoid each other

A fascinating paper by Simona Cafazzo, Sarah Marshall-Pescini, Martina Lazzaroni, Zsófia Virányi, Friederike Range in the Royal Society Open Science

Highly cooperative social species are expected to engage in frequent reconciliation following conflicts in order to maintain pack cohesiveness and preserve future cooperation. By contrast, in social species with low reliance on cooperation, reconciliation is expected to be less frequent. Here, we investigate the pattern of reconciliation in four captive wolf packs and four captive dog packs. We provide evidence for reconciliation in captive wolves, which are highly dependent on cooperation between pack members, while domestic dogs, which rely on conspecific cooperation less than wolves, avoided interacting with their partners after conflicts. Occurrence, intensity, latency, duration and initiation of wolf reconciliations appeared to vary as a consequence of a compromise between the costs (e.g. risk of further aggression) and the benefits (e.g. restoring relationship with opponents) of such interactions. Our results are in line with previous findings on various wolf packs living under different social and ecological conditions, suggesting that reconciliation is an important strategy for maintaining functional relationships and pack cohesiveness. However, current results on dogs are in contrast to the only other study showing that reconciliation can occur also in this species. Therefore, the occurrence of reconciliation in dogs may be influenced by social and environmental conditions more than in wolves. Which factors promote and modulate reconciliation in dogs needs to be further investigated.


Understanding Mathematics

Question on a LinkedIn discussion group:

Which way do you learn and understand Mathematics in general? -> From the “forest” to the “tree” (-> first the Big Picture and then going into the details) or vice versa (-> seeing the Big Picture after learning and understanding some details) and explain why.

My answer:

In learning mathematics, it definitely starts from a tree to a forest. I think only professional research mathematicians, or exceptional teachers of mathematics, can really think from a “forest” to a “tree”. I recall a recent conversation with my former MSc and PhD student who now works, in a senior position, in software development for a serious, and widely known, Internet company. I asked her: “Does it help you that you have a PhD in mathematics”. Her response: “When they have a project too big to handle, they come to me and I cut it in smaller more manageable parts”. This is thinking from top down. It remains a rare skill, hard to teach, and hard to learn. And the issue is of critical importance not only for mathematics, but for other walks of life, too. Just a few weeks ago I had a conversation about that with a lecturer of architecture (and she was also a very successful practicing architect).


Why it matters that so many people became innumerate

BBC Capital published this post, Why it matters if we become innumerate. The article says:

“What we all need in daily life is quite simple maths,” says Mike Ellicock, chief executive of the charity National Numeracy. “But we also need a conceptual understanding applied to complex situations.” In essence, this understanding applies to a broad range of mathematical information that may be intricate, abstract, or embedded in unfamiliar contexts.

For instance, you might need to calculate the true cost of buying versus hiring a car; whether to use award points or money to buy an airline ticket; or how to adjust a recipe to feed six people instead of four.

In the first two examples, you deal with businesses which are not much interested in you being numerate. I have already said on a number of occasions that

If banks and insurance companies were interested in having numerate customers – as they occasionally claim – we would witness the golden age of school mathematics: fully funded, enjoying cross-party political support, promoted and popularised by the best advertising companies in all forms of mass and social media. But they are not; banks and insurance companies need numerate workforce – but even more so they need innumerate customers. 25 years ago in the West, the benchmark of arithmetic competence at a consumer level was the ability to balance a chequebook. Nowadays, bank customers can instantly get full information about the state of their accounts from an app on a mobile phone together with a timely and tailored to individual circumstances advice on the range of recommended financial products. This kind service can be described in a logically equivalent form: a bank can instantly exploit the customer’s vulnerability.

Growing innumeracy is a quite alarming socio-economic phenomenon, and it has very deep roots. Read mor on that in my papers:

Mathematics for makers and mathematics for users, bit.ly/2qYHtst

Calling a spade a spade: Mathematics in the new pattern of division of labour, goo.gl/TT6ncO


Why does algebra have letters in sums?

My answer to a question on Quora:

Why does algebra have letters in sums?

One should not underestimate the influence of François Viète who was the first to use algebraic notation (letters) not only for unknowns but also for parameters (knowns) in a problem. He also used, as Wiki states, “simplification of equations by the substitution of new quantities having a certain connection with the primitive unknown quantities”. Importantly, Viète is the first cryptographer and cryptanalist know to us by name. His decryption of intercepted diplomatic correspondence had direct effect on European politics of his time. A really juicy bit from the Wiki:

In 1590, Viète discovered the key to a Spanish cipher, consisting of more than 500 characters, and this meant that all dispatches in that language which fell into the hands of the French could be easily read.

Henry IV published a letter from Commander Moreo to the king of Spain. The contents of this letter, read by Viète, revealed that the head of the League in France, the Duke of Mayenne, planned to become king in place of Henry IV. This publication led to the settlement of the Wars of Religion. The king of Spain accused Viète of having used magical powers.

At that time, encryption of texts mostly used substitution ciphers, and the idea of substitution of letters for numbers should be very natural for Vieta.

I modestly suggest that teachers could perhaps use this idea: teaching primary school children some basic substitution ciphers: it is fun, it is a natural spelling exercise, and, I believe, a good propaedeutic for later study of algebra and computer coding.


No intrinsic gender differences in children’s earliest numerical abilities

Alyssa J. Kersey, Emily J. Braham, Kelsey D. Csumitta, Melissa E. Libertus & Jessica F. Cantlon. No intrinsic gender differences in children’s earliest numerical abilities. npj Science of Learning  3, Article number: 12 (2018).


Recent public discussions have suggested that the underepresentation of women in science and mathematics careers can be traced back to intrinsic differences in aptitude. However, true gender differences are difficult to assess because sociocultural influences enter at an early point in childhood. If these claims of intrinsic differences are true, then gender differences in quantitative and mathematical abilities should emerge early in human development. We examined cross-sectional gender differences in mathematical cognition from over 500 children aged 6 months to 8 years by compiling data from five published studies with unpublished data from longitudinal records. We targeted three key milestones of numerical development: numerosity perception, culturally trained counting, and formal and informal elementary mathematics concepts. In addition to testing for statistical differences between boys’ and girls’ mean performance and variability, we also tested for statistical equivalence between boys’ and girls’ performance. Across all stages of numerical development, analyses consistently revealed that boys and girls do not differ in early quantitative and mathematical ability. These findings indicate that boys and girls are equally equipped to reason about mathematics during early childhood.